Pringsheim's Theorem

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Theorem

Let $f$ be a holomorphic function defined on a unit disc centered at the origin of the complex plane and is denoted by its Taylor series:

$\map f z = \ds \sum_{n \mathop = 0}^{\infty} c_n z^n$

Let:

$(1): \quad \forall n \ge 0: c_n \ge 0$

$(2): \quad$ the radius of convergence of the Taylor series of function $f$ is $1$.

Then $z = 1$ is an isolated singularity of $f$.

Proof

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Source of Name

This entry was named for Alfred Pringsheim.